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Abstract: In this paper, we investigate two-species Lotka–Volterra delayed stochastic predator–prey systems, with and without pollution, denoted by and , respectively. We show that there exists a unique non-negative solution in each system that is permanent in time average under certain conditions. Moreover, the non-permanence of model is studied. Finally, computer simulations are carried out to verify our results. [Copyright &y& Elsevier]

Abstract: This paper analyzes the asymptotic behaviors of the stochastic population model with the Allee effect. We apply the Feller’s test to obtain the criteria of the asymptotic behaviors for this model and compare our results with those obtained by Lyapunov functions. At last, the numerical simulations conform our results. [Copyright &y& Elsevier]

Abstract: A stochastic ratio-dependent predator–prey model is investigated in this paper. By the comparison theorem of stochastic equations and Itô’s formula, we obtain the global existence of a positive unique solution of the ratio-dependent model. Besides, a condition for species to be extinct is given and a persistent condition is established. We also conclude that both the prey population and the ratio-dependent function are stable in time average. In the end, numerical simulations are carried out to confirm our findings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique positive solution to the system with positive initial value, and mainly investigate the long time behavior of the system. Condition for the system to be extinct is given and persistent condition is established. At last, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized non-autonomous logistic equation , where is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164–172], the authors show that has a unique positive T-periodic solution provided , and are continuous T-periodic functions, , and . We show that this equation is stochastically permanent and the solution is globally attractive provided , and are continuous T-periodic functions, , and . By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized nonautonomous logistic equation where is 1-dimensional standard Brownian motion. We show that has a unique positive T-periodic solution provided , , and are continuous T-periodic functions, , and . [Copyright &y& Elsevier]